At the atomic-scale, the electron volt and nanometer

Estimate the peak wavelength for radiation from (a) ice at 273K, (ib) a floodlamp at 3500 K, (c) helium at 4.2 K, (d) for the universe at T = 2.725 K, assuming blackbody emission. In what region of the EM spectrum is each?

How hot is metal being welded if it radiates most strongly at 460 nm?

An HC1 molecule vibrates with a natural frequency of 8.1 X 1013 Hz. What is the difference in energy (in joules and electron volts) between successive values of the oscillation energy?

Estimate the peak wavelength of light issuing from the pupil of the human eye (which approximates a blackbody) assuming normal body temperature.

Plancks radiation law is given by:where /(A, T) is the rate energy is radiated per unit surface area per unit wavelength interval at wavelength A and Kelvin temperature T. (a) Show that Wiens displacement law follows from this relationship, (b) Determine the value of h from the experimental value of APT given in the text. [You may want to use graphing techniques.] (c) Derive the r 4 dependence of the rate at which energy is radiated (as in the Stefan-Boltzmann law, Eq. 19-17), by integrating Plancks formula over all wavelengths; that is, show that

What is the energy of photons (in joules) emitted by a 104.1-MHz FM radio station?

What is the energy range (in joules and eV) of photons in the visible spectrum, of wavelength 410 nm to 750 nm?

A typical gamma ray emitted from a nucleus during radioactive decay may have an energy of 380 keV. What is its wavelength? Would we expect significant diffraction of this type of light when it passes through an everyday opening, such as a door?

About 0.1 eV is required to break a hydrogen bond in a protein molecule. Calculate the minimum frequency and maximum wavelength of a photon that can accomplish this.

Calculate the momentum of a photon of yellow light of wavelength 6.20 X 10 7 m.

What minimum frequency of light is needed to eject electrons from a metal whose work function is 4.8 X 1 0 19 J?

What is the longest wavelength of light that will emit electrons from a metal whose work function is 3.70 eV?

What wavelength photon would have the same energy as a 145-gram baseball moving 30.0 m/s?

The human eye can respond to as little as 10-18 J of light energy. For a wavelength at the peak of visual sensitivity, 550 nm, how many photons lead to an observable flash?

The work functions for sodium, cesium, copper, and iron are 2.3, 2.1, 4.7, and 4.5 eV, respectively. Which of these metals will not emit electrons when visible light shines on it?

In a photoelectric-effect experiment it is observed that no current flows unless the wavelength is less than 520 nm. (a) What is the work function of this material? (b) What is the stopping voltage required if light of wavelength 470 nm is used?

What is the maximum kinetic energy of electrons ejected from barium (W0 = 2.48 eV) when illuminated by white light, A = 410 to 750 nm?

Barium has a work function of 2.48 eV. What is the maximum kinetic energy of electrons if the metal is illuminated by UV light of wavelength 365 nm? What is their speed?

When UV light of wavelength 285 nm falls on a metal surface, the maximum kinetic energy of emitted electrons is 1.70 eV. What is the work function of the metal?

The threshold wavelength for emission of electrons from a given surface is 320 nm. What will be the maximum kinetic energy of ejected electrons when the wavelength is changed to (a) 280 nm, (b) 360 nm?

When 230-nm light falls on a metal, the current through a photoelectric circuit (Fig. 37-4) is brought to zero at a stopping voltage of 1.84 V. What is the work function of the metal?

A certain type of film is sensitive only to light whose wavelength is less than 630 nm. What is the energy (eV and kcal/mol) needed for the chemical reaction to occur which causes the film to change?

The range of visible light wavelengths extends from about 410 nm to 750 nm. (a) Estimate the minimum energy (eV) necessary to initiate the chemical process on the retina that is responsible for vision, (b) Speculate as to why, at the other end of the visible range, there is a threshold photon energy beyond which the eye registers no sensation of sight. Determine this threshold photon energy (eV).

In a photoelectric experiment using a clean sodium surface, the maximum energy of the emitted electrons was measured for a number of different incident frequencies, with the following results.Frequency (x 1014 Hz) Energy (eV) 11.8 2.60 10.6 2.11 9.9 1.81 9.1 1.47 8.2 1.10 6.9 0.57 Plot the graph of these results and find: (a) Plancks constant; (b) the cutoff frequency of sodium; (c) the work function.

A photomultiplier tube (a very sensitive light sensor), is based on the photoelectric effect: incident photons strike a metal surface and the resulting ejected electrons are collected. By counting the number of collected electrons, the number of incident photons (i.e., the incident light intensity) can be determined. (a) If a photomultiplier tube is to respond properly for incident wavelengths throughout the visible range (410 nm to 750 nm), what is the maximum value for the work function W0 (eV) of its metal surface? (b) If W0 for its metal surface is above a certain threshold value, the photomultiplier will only function for incident ultraviolet wavelengths and be unresponsive to visible light. Determine this threshold value (eV).

A group of atoms is confined to a very small (point-like) volume in a laser-based atom trap. The incident laser light causes each atom to emit 1.0 X 106 photons of wavelength 780 nm every second. A sensor of area 1.0 cm2 measures the light intensity emanating from the trap to be 1.6 nW when placed 25 cm away from the trapped atoms. Assuming each atom emits photons with equal probability in all directions, determine the number of trapped atoms.

Assume light of wavelength A is incident on a metal surface, whose work function is known precisely (i.e., its uncertainty is better than 0.1% and can be ignored). Show that if the stopping voltage can be determined to an accuracy of A ^ , the fractional uncertainty (magnitude) in wavelength is Determine this fractional uncertainty if = 0.01 V and

A high-frequency photon is scattered off of an electron and experiences a change of wavelength of 1.5 X 10-4 nm. At what angle must a detector be placed to detect the scattered photon (relative to the direction of the incoming photon)?

Determine the Compton wavelength for (a) an electron, (b) a proton, (c) Show that if a photon has wavelength equal to the Compton wavelength of a particle, the photons energy is equal to the rest energy of the particle.

X-rays of wavelength A = 0.120 nm are scattered from carbon. What is the expected Compton wavelength shift for photons detected at angles (relative to the incident beam) of exactly {a) 60, (b) 90, (c) 180?

In the Compton effect, determine the ratio (AA/A) of the maximum change AA in a photons wavelength to the photons initial wavelength A, if the photon is (a) a visiblelight photon with A = 550 nm, (b) an X-ray photon with A = 0.10 nm.

A 1.0-MeV gamma-ray photon undergoes a sequence of Compton-scattering events. If the photon is scattered at an angle of 0.50 in each event, estimate the number of events required to convert the photon into a visible-light photon with wavelength 555 nm. You can use an expansion for small 0; see Appendix A. [Gamma rays created near the center of the Sun are transformed to visible wavelengths as they travel to the Suns surface through a sequence of small-angle Compton scattering events.]

In the Compton effect, a 0.160-nm photon strikes a free electron in a head-on collision and knocks it into the forward direction. The rebounding photon recoils directly backward. Use conservation of (relativistic) energy and momentum to determine (a) the kinetic energy of the electron, and (b) the wavelength of the recoiling photon. Use Eq. 37-5, but not Eq. 37-6.

In the Compton effect (see Fig. 37-7), use the relativistic equations for conservation of energy and of linear momentum to show that the Compton shift in wavelength is given by Eq. 37-6.

How much total kinetic energy will an electron-positron pair have if produced by a 2.67-MeV photon?

What is the longest wavelength photon that could produce a proton-antiproton pair? (Each has a mass of 1.67 X 1027kg.)

What is the minimum photon energy needed to produce a /jl+ - pair? The mass of each /jl (muon) is 207 times the mass of an electron. What is the wavelength of such a photon?

An electron and a positron, each moving at 2.0 X 105 m/s, collide head on, disappear, and produce two photons moving in opposite directions, each with the same energy and momentum. Determine the energy and momentum of each photon.

A gamma-ray photon produces an electron and a positron, each with a kinetic energy of 375 keV. Determine the energy and wavelength of the photon.

Calculate the wavelength of a 0.23-kg ball traveling at 0.10 m/s.

What is the wavelength of a neutron (ra = 1.67 X 10_27kg) traveling at 8.5 X 104m/s?

Through how many volts of potential difference must an electron be accelerated to achieve a wavelength of 0.21 nm?

What is the theoretical limit of resolution for an electron microscope whose electrons are accelerated through 85 kV? (Relativistic formulas should be used.)

The speed of an electron in a particle accelerator is 0.98c. Find its de Broglie wavelength. (Use relativistic momentum.)

Calculate the ratio of the kinetic energy of an electron to that of a proton if their wavelengths are equal. Assume that the speeds are nonrelativistic.

Neutrons can be used in diffraction experiments to probe the lattice structure of crystalline solids. Since the neutrons wavelength needs to be on the order of the spacing between atoms in the lattice, about 0.3 nm, what should the speed of the neutrons be?

An electron has a de Broglie wavelength A = 6.0 X IO-10 m. (a) What is its momentum? (b) What is its speed? (c) What voltage was needed to accelerate it to this speed?

What is the wavelength of an electron of energy (a) 20 eV, (b) 200 eV, (c) 2.0 keV?

Show that if an electron and a proton have the same nonrelativistic kinetic energy, the proton has the shorter wavelength.

Calculate the de Broglie wavelength of an electron in a TV picture tube if it is accelerated by 33,000 V. Is it relativistic? How does its wavelength compare to the size of the neck of the tube, typically 5 cm? Do we have to worry about diffraction problems blurring our picture on the screen?

After passing through two slits separated by a distance of 3.0 /mi, a beam of electrons creates an interference pattern with its second-order maximum at an angle of 55. Find the speed of the electrons in this beam.

What voltage is needed to produce electron wavelengths of 0.28 nm? (Assume that the electrons are nonrelativistic.)

Electrons are accelerated by 3450 V in an electron microscope. Estimate the maximum possible resolution of the microscope.

For the three hydrogen transitions indicated below, with n being the initial state and n' being the final state, is the transition an absorption or an emission? Which is higher, the initial state energy or the final state energy of the atom? Finally, which of these transitions involves the largest energy photon? (a) n = 1, n' = 3; (b) n = 6, n' = 2; (c) n = 4, n' = 5.

How much energy is needed to ionize a hydrogen atom in the n = 3 state?

(a) Determine the wavelength of the second Balmer line (n = 4 to n = 2 transition) using Fig. 37-26. Determine likewise (b) the wavelength of the third Lyman line and (c) the wavelength of the first Balmer line.

Calculate the ionization energy of doubly ionized lithium, Li2+, which has Z = 3.

Evaluate the Rydberg constant R using the Bohr model (compare Eqs. 37-8 and 37-15) and show that its value is R = 1.0974 X 107 m_1.

What is the longest wavelength light capable of ionizing a hydrogen atom in the ground state?

In the Sun, an ionized helium (He+) atom makes a transition from the n = 5 state to the n = 2 state, emitting a photon. Can that photon be absorbed by hydrogen atoms present in the Sun? If so, between what energy states will the hydrogen atom transition occur?

What wavelength photon would be required to ionize a hydrogen atom in the ground state and give the ejected electron a kinetic energy of 20.0 eV?

For what maximum kinetic energy is a collision between an electron and a hydrogen atom in its ground state definitely elastic?

Construct the energy-level diagram for the He+ ion (like Fig. 37-26).

Construct the energy-level diagram (like Fig. 37-26) for doubly ionized lithium, Li2+.

Determine the electrostatic potential energy and the kinetic energy of an electron in the ground state of the hydrogen atom.

An excited hydrogen atom could, in principle, have a diameter of 0.10 mm. What would be the value of n for a Bohr orbit of this size? What would its energy be?

Is the use of nonrelativistic formulas justified in the Bohr atom? To check, calculate the electrons velocity, v, in terms of c, for the ground state of hydrogen, and then calculate \ / l - v2lc2.

A hydrogen atom has an angular momentum of 5.273 X 10-34 kg m2/s. According to the Bohr model, what is the energy (eV) associated with this state?

Assume hydrogen atoms in a gas are initially in their ground state. If free electrons with kinetic energy 12.75 eV collide with these atoms, what photon wavelengths will be emitted by the gas?

Suppose an electron was bound to a proton, as in the hydrogen atom, but by the gravitational force rather than by the electric force. What would be the radius, and energy, of the first Bohr orbit?

Correspondence principle: Show that for large values of n, the difference in radius Ar between two adjacent orbits (with quantum numbers n and n - 1) is given by Ar = rn - rn- i > n so A r/rn > 0 as n oo in accordance with the correspondence principle. [Note that we can check the correspondence principle by either considering large values of n (n oo) or by letting h 0. Are these equivalent?]

If a 75-W lightbulb emits 3.0% of the input energy as visible light (average wavelength 550 nm) uniformly in all directions, estimate how many photons per second of visible light will strike the pupil (4.0 mm diameter) of the eye of an observer 250 m away.

At low temperatures, nearly all the atoms in hydrogen gas will be in the ground state. What minimum frequency photon is needed if the photoelectric effect is to be observed?

A beam of 125-eV electrons is scattered from a crystal, as in X-ray diffraction, and a first-order peak is observed at 6 = 38. What is the spacing between planes in the diffracting crystal? (See Section 35-10.)

A microwave oven produces electromagnetic radiation at A = 12.2 cm and produces a power of 860 W. Calculate the number of microwave photons produced by the microwave oven each second.

Sunlight reaching the Earth has an intensity of about 1350 W/m2. Estimate how many photons per square meter per second this represents. Take the average wavelength to be 550 nm.

A beam of red laser light (A = 633 nm) hits a black wall and is fully absorbed. If this light exerts a total force F = 6.5 nN on the wall, how many photons per second are hitting the wall?

The Big Bang theory states that the beginning of the universe was accompanied by a huge burst of photons. Those photons are still present today and make up the so-called cosmic microwave background radiation. The universe radiates like a blackbody with a temperature of about 2.7 K. Calculate the peak wavelength of this radiation.

An electron and a positron collide head on, annihilate, and create two 0.755-MeV photons traveling in opposite directions. What were the initial kinetic energies of electron and positron?

By what potential difference must (a) a proton (m = 1.67 X 10-27 kg), and (b) an electron (m = 9.11 X 10-31 kg), be accelerated to have a wavelength A = 6.0 X 10-12 m?

In some of Rutherfords experiments (Fig. 37-17) the a particles (mass = 6.64 X 10 27 kg) had a kinetic energy of 4.8 MeV. How close could they get to the center of a silver nucleus (charge = +47e)? Ignore the recoil motion of the nucleus.

Show that the magnitude of the electrostatic potential energy of an electron in any Bohr orbit of a hydrogen atom is twice the magnitude of its kinetic energy in that orbit.

Calculate the ratio of the gravitational force to the electric force for the electron in a hydrogen atom. Can the gravitational force be safely ignored?

Electrons accelerated by a potential difference of 12.3 V pass through a gas of hydrogen atoms at room temperature. What wavelengths of light will be emitted?

In a particular photoelectric experiment, a stopping potential of 2.70 V is measured when ultraviolet light of wavelength 380 nm is incident on the metal. If blue light of wavelength 440 nm is used, what is the new stopping potential?

In an X-ray tube (see Fig. 35-26 and discussion in Section 35-10), the high voltage between filament and target is V. After being accelerated through this voltage, an electron strikes the target where it is decelerated (by positively charged nuclei) and in the process one or more X-ray photons are emitted, (a) Show that the photon of shortest wavelength will have (b) What is the shortest wavelength of X-ray emitted when accelerated electrons strike the face of a 33-kV television picture tube?

The intensity of the Suns light in the vicinity of Earth is about 1350 W/m2. Imagine a spacecraft with a mirrored square sail of dimension 1.0 km. Estimate how much thrust (in newtons) this craft will experience due to collisions with the Suns photons. [Hint: Assume the photons bounce perpendicularly off the sail with no change in the magnitude of their momentum.]

Photons of energy 9.0 eV are incident on a metal. It is found that current flows from the metal until a stopping potential of 4.0 V is applied. If the wavelength of the incident photons is doubled, what is the maximum kinetic energy of the ejected electrons? What would happen if the wavelength of the incident photons was tripled?

Light of wavelength 360 nm strikes a metal whose work function is 2.4 eV. What is the shortest de Broglie wavelength for the electrons that are produced as photoelectrons?

Visible light incident on a diffraction grating with slit spacing of 0.012 mm has the first maximum at an angle of 3.5 from the central peak. If electrons could be diffracted by the same grating, what electron velocity would produce the same diffraction pattern as the visible light?

(a) Suppose an unknown element has an absorption spectrum with lines corresponding to 2.5, 4.7, and 5.1 eV above its ground state, and an ionization energy of 11.5 eV. Draw an energy level diagram for this element. (b) If a 5.1-eV photon is absorbed by an atom of this substance, in which state was the atom before absorbing the photon? What will be the energies of the photons that can subsequently be emitted by this atom?

Light of wavelength 424 nm falls on a metal which has a work function of 2.28 eV. (a) How much voltage should be applied to bring the current to zero? (b) What is the maximum speed of the emitted electrons? (c) What is the de Broglie wavelength of these electrons?

Apply Bohrs assumptions to the Earth-Moon system to calculate the allowed energies and radii of motion. Given the known distance between Earth and the Moon, is the quantization of the energy and radius apparent?

Show that the wavelength of a particle of mass ra with kinetic energy K is given by the relativistic formula A = h c /y /K 2 + 2 mc2K .

A small flashlight is rated at 3.0 W. As the light leaves the flashlight in one direction, a reaction force is exerted on the flashlight in the opposite direction. Estimate the size of this reaction force.

At the atomic-scale, the electron volt and nanometer are well-suited units for energy and distance, respectively. (a) Show that the energy E in eV of a photon, whose wavelength A is in nm, is given by ^ _ 1240 eV nm A (nm)(b) How much energy (eV) does a 650-nm photon have?

Three fundamental constants of naturethe gravitational constant G, Plancks constant h, and the speed of light c have the dimensions of [L3/M T 2], [ML2/T], and [L/T], respectively, (a) Find the mathematical combination of these fundamental constants that has the dimension of time. This combination is called the Planck time tP and is thought to be the earliest time, after the creation of the universe, at which the currently known laws of physics can be applied. (b) Determine the numerical value of tP. (c) Find the mathematical combination of these fundamental constants that has the dimension of length. This combination is called the Planck length AP and is thought to be the smallest length over which the currently known laws of physics can be applied, (d) Determine the numerical value of AP.

Imagine a free particle of mass ra bouncing back and forth between two perfectly reflecting walls, separated by distance L Imagine that the two oppositely directed matter waves associated with this particle interfere to create a standing wave with a node at each of the walls. Show that the ground state (first harmonic) and first excited state (second harmonic) have (non-relativistic) kinetic energies A2/8ra2 and /z2/2mf2, respectively.

(a) A rubidium atom (ra = 85 u) is at rest with one electron in an excited energy level. When the electron jumps to the ground state, the atom emits a photon of wavelength A = 780 nm. Determine the resulting (nonrelativistic) recoil speed v of the atom. (b) The recoil speed sets the lower limit on the temperature to which an ideal gas of rubidium atoms can be cooled in a laser-based atom trap. Using the kinetic theory of gases (Chapter 18), estimate this lowest achievable temperature.

A rubidium atom (atomic mass 85) is initially at room temperature and has a velocity v = 290 m/s due to its thermal motion. Consider the absorption of photons by this atom from a laser beam of wavelength A = 780 nm. Assume the rubidium atoms initial velocity v is directed into the laser beam (the photons are moving right and the atom is moving left) and that the atom absorbs a new photon every 25 ns. How long will it take for this process to completely stop (cool) the rubidium atom? [Note: a more detailed analysis predicts that the atom can be slowed to about 1 cm/s by this light absorption process, but it cannot be completely stopped.]

(a) Graph Plancks radiation formula (top of page 989) as a function of wavelength from A = 20 nm to 2000 nm in 20 nm steps for two lightbulb filaments, one at 2700 K and the other at 3300 K. Plot both curves on the same set of axes. (b) Approximately how much more intense is the visible light from the hotter bulb? Use numerical integration.

Estimate what % of emitted sunlight energy is in the visible range. Use Plancks radiation formula (top of page 989) and numerical integration.

Potassium has one of the lowest work functions of all metals and so is useful in photoelectric devices using visible light. Light from a source is incident on a potassium surface. Data for the stopping voltage V0 as a function of wavelength A is shown below, (a) Explain why a graph of V0 vs. 1/A is expected to yield a straight line. What are the theoretical expectations for the slope and the ^-intercept of this line? (b ) Using the data below, graph Vq vs. 1/A and show that a straight-line plot does indeed result. Determine the slope a and ^-intercept b of this line. Using your values for a and b, determine (c) potassiums work function (eV) and (d) Plancks constant h (J-s). A (/mi) 0^00 0430 0460 0490 0520~ V0 (V) 0.803 0.578 0.402 0.229 0.083